Legendre spectral-collocation method for Volterra integral differential equations with nonvanishing delay (Q394239)

The Volterra integral differential equations with nonvanishing delay considered in this paper are as follows: NEWLINE\[NEWLINE y'(t)=a(t)y(t)+b(t)y(\theta(t))+g(t)+\int_{0}^{t}K_1(t,s)y(s)ds+\int_{0}^{\theta(t)}K_2(t,s)y(s)ds, NEWLINE\]NEWLINE NEWLINE\[NEWLINE t\in(0,T], \quad y(t)=\varphi(t),\quad t\in[\theta(0),0]. NEWLINE\]NEWLINE It is assumed that all the functions in this equation possess continuous derivatives of at least order \(m\geq 1\) on their respective domains.NEWLINENEWLINEThe delay function \(\theta(t)\) satisfies the following conditions: NEWLINE\[NEWLINE \theta(t):=t-\tau(t),\quad \tau\in C^m([0,T]), NEWLINE\]NEWLINE NEWLINE\[NEWLINE \tau(t)\geq\tau_0>0 \text{ for all } t\in[0,T], NEWLINE\]NEWLINE NEWLINE\[NEWLINE \theta \text{ is strictly increasing on } [0,T]. NEWLINE\]NEWLINENEWLINENEWLINEThe main purpose of this paper is to propose the Legendre spectral-collocation method to solve the Volterra integral differential equations with nonvanishing delay. The authors divide the definition domain of the solution into several subintervals where the solution is sufficiently smooth. Then the spectral-collocation method for these equations in each subinterval can be used. A convergence analysis for this method is provided, which shows that the numerical errors decay exponentially. Numerical examples are presented to confirm these theoretical results.